Let $ p \in \mathbb {Z}[X] $ and of fifth degree with only the terms for $x^5$, $x^4$ and $x^3$ known, the terms for $x^2$, $x$ and $1$ are not known. $$ p=x^5  5x^4  35x^3 + \cdots $$ The ( five ) roots of $p $ form an arithmetic sequence. Find the roots of $p $.
Wednesday, October 22, 2014
Monday, September 15, 2014
Saturday, August 2, 2014
A Five By Five Magic Square
A few years ago I wanted to find a 5 by five magic square. I gave up on it. See this post for the results I came up with in 2011. I used the wrong tools, I can see that clearly now.
Yesterday and today I worked on it. Again without finding a solution. This morning I thought "Should I spend another day on it? What am I doing with my time? I am not going to solve this one ( either )". But problems you can't solve become little traumas in your subconscious. From time to time they remind you that you weren't able to crack them. It hurts. Probably more than you can imagine.
There was one thing I could add to my search algorithm, I decided to add that and then stop. I had only one diagonal left that didn't add up to 65. Anyway, the last thing worked and I found one. One hurting trauma less.
$$
\begin{array}{ccccc}
2 & 25 & 5 & 20 & 13 \\
18 & 4 & 17 & 7 & 19 \\
8 & 24 & 22 & 10 & 1 \\
23 & 9 & 6 & 16 & 11 \\
14 & 3 & 15 & 12 & 21
\end{array}
$$
The integers $1,2 \dots 25$ are laid out in a 5 by 5 matrix such that the rows, columns and the two diagonals total to 65.
Now the real fun begins: finding bigger ones, special ones maybe, and analyzing and optimizing my method. Where are the limits? And questions, many questions like how many different 5 by 5s are there? Is there one with the number 13 in the centre?
Yesterday and today I worked on it. Again without finding a solution. This morning I thought "Should I spend another day on it? What am I doing with my time? I am not going to solve this one ( either )". But problems you can't solve become little traumas in your subconscious. From time to time they remind you that you weren't able to crack them. It hurts. Probably more than you can imagine.
There was one thing I could add to my search algorithm, I decided to add that and then stop. I had only one diagonal left that didn't add up to 65. Anyway, the last thing worked and I found one. One hurting trauma less.
$$
\begin{array}{ccccc}
2 & 25 & 5 & 20 & 13 \\
18 & 4 & 17 & 7 & 19 \\
8 & 24 & 22 & 10 & 1 \\
23 & 9 & 6 & 16 & 11 \\
14 & 3 & 15 & 12 & 21
\end{array}
$$
The integers $1,2 \dots 25$ are laid out in a 5 by 5 matrix such that the rows, columns and the two diagonals total to 65.
Now the real fun begins: finding bigger ones, special ones maybe, and analyzing and optimizing my method. Where are the limits? And questions, many questions like how many different 5 by 5s are there? Is there one with the number 13 in the centre?
Sunday, May 18, 2014
Explained: Mathematical Virus type MV/C
ARE YOU INFECTED WITH MV/C ?
What is MV
A mathematical virus (MV) is a preconception about the structure, function or method of mathematics which impairs one's ability to do mathematics.
What is MV/C?
MV/C is a mathematical virus which is easily diagnosed, the infected suffer from the delusion that "Coordinates are essential to calculations". Physicists and engineers are especially susceptible to this virus, because most of their textbooks are infected, and infected teachers pass it on to their students.
The first MV was discovered by David Hestenes, a theoretical physicist, best known as chief architect of geometric algebra as a unified language for mathematics and physics.
References
David Hestenes, Mathematical VirusesSunday, December 15, 2013
( Progress on the ) tiling printer.
Unicolor2D graphics can't be spectaculair, let alone breathtaking. Unless you are in the know. Let me explain. I am now able to  almost  print tilings automatically from just a string of numbers as I wrote about in a previous post.
Not spectaculair at all, it's just a collection of 8 tilings containing 72 polygons in total. But if I change ( 4,2 ) by (10,5) and add a colorfunction the program creates immediately the following graphic.
Both are, in fact, representations of the same sequence.{6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4} but the variations from this sequence alone are endless. The program creates a graphic for every sequence input. Not all graphics will be tilings of the plane by definition. The following ( major ) step is to let the program reject sequences that do not lead to tilings of the plane.
Tne next part will include an implentation of a 'Point in Polygon' predicate. Many of those exist and have been implemented for many languages. That could be a topic for a more indepth post next time.
Not spectaculair at all, it's just a collection of 8 tilings containing 72 polygons in total. But if I change ( 4,2 ) by (10,5) and add a colorfunction the program creates immediately the following graphic.
Both are, in fact, representations of the same sequence.{6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4} but the variations from this sequence alone are endless. The program creates a graphic for every sequence input. Not all graphics will be tilings of the plane by definition. The following ( major ) step is to let the program reject sequences that do not lead to tilings of the plane.
Tne next part will include an implentation of a 'Point in Polygon' predicate. Many of those exist and have been implemented for many languages. That could be a topic for a more indepth post next time.
Sunday, December 8, 2013
Challenging the eternal truth of mathematics
I learned that 0,999... = 1. I believe it was in M381 that I learned to prove it. The proof is quite simple actually.
Let
x = 0,999..., multiply both sides with 10
10x = 9,999..., now subtract x from 10x
9x = 9, and we have our result
x = 1.
What I like about mathematics is that it is timeless, i.e. we can still read math books of hundreds of years old and still learn things from them. That's not an advisable strategy hfor any other science than mathematics. Of course, new branches of mathematics appear, new discoveries are made, but they don't invalidate the truths of the past.
Today however I found someone who actually is challenging some of the established truths in mathematics, like for example that 0,999... = 1. I don't think that he is a crackpot, although I have no doubt that the mathematical establishment, professors who are 'safe' by all means, will call him like that.
The man is arrogant though, he calls the proof above, 'juvenile' for example.
Who is he, what are his ideas and how did he disproof that 1 = 0,999? The links below will help you abshereing these questions.
 The New Calculus  The first rigorous formulation of calculus in history.
 Proof that 0.999 not equal 1.pdf
Let
x = 0,999..., multiply both sides with 10
10x = 9,999..., now subtract x from 10x
9x = 9, and we have our result
x = 1.
What I like about mathematics is that it is timeless, i.e. we can still read math books of hundreds of years old and still learn things from them. That's not an advisable strategy hfor any other science than mathematics. Of course, new branches of mathematics appear, new discoveries are made, but they don't invalidate the truths of the past.
Today however I found someone who actually is challenging some of the established truths in mathematics, like for example that 0,999... = 1. I don't think that he is a crackpot, although I have no doubt that the mathematical establishment, professors who are 'safe' by all means, will call him like that.
The man is arrogant though, he calls the proof above, 'juvenile' for example.
Who is he, what are his ideas and how did he disproof that 1 = 0,999? The links below will help you abshereing these questions.
 The New Calculus  The first rigorous formulation of calculus in history.
 Proof that 0.999 not equal 1.pdf
Some progress...
If you read the previous posts on my tiling printing algorithms you'll understand the reason for this post: I made some progress that unraveled some serious knots in my stomach.
{ 4, 2, 4 } >
{ 6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4 } >
The second white polygon is made from the following points :
\begin{array}{cc}
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
0 & 1 \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{1}{2} \left(1+\sqrt{3}\right) & \frac{1}{2} \left(1+\sqrt{3}\right) \\
\frac{1}{2} \left(1+\sqrt{3}\right) & \frac{1}{2} \left(3+\sqrt{3}\right) \\
\frac{\sqrt{3}}{2} & \frac{3}{2}+\sqrt{3} \\
0 & 1+\sqrt{3} \\
\frac{\sqrt{3}}{2} & \frac{3}{2}+\sqrt{3} \\
\frac{1}{2} \left(1\sqrt{3}\right) & \frac{1}{2} \left(3+\sqrt{3}\right) \\
\frac{1}{2} \left(1\sqrt{3}\right) & \frac{1}{2} \left(1+\sqrt{3}\right) \\
\end{array}
Ready to enter the next level of the problem. ;)
{ 4, 2, 4 } >
{ 6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4 } >
The second white polygon is made from the following points :
\begin{array}{cc}
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
0 & 1 \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{1}{2} \left(1+\sqrt{3}\right) & \frac{1}{2} \left(1+\sqrt{3}\right) \\
\frac{1}{2} \left(1+\sqrt{3}\right) & \frac{1}{2} \left(3+\sqrt{3}\right) \\
\frac{\sqrt{3}}{2} & \frac{3}{2}+\sqrt{3} \\
0 & 1+\sqrt{3} \\
\frac{\sqrt{3}}{2} & \frac{3}{2}+\sqrt{3} \\
\frac{1}{2} \left(1\sqrt{3}\right) & \frac{1}{2} \left(3+\sqrt{3}\right) \\
\frac{1}{2} \left(1\sqrt{3}\right) & \frac{1}{2} \left(1+\sqrt{3}\right) \\
\end{array}
Ready to enter the next level of the problem. ;)
Saturday, November 30, 2013
Organizing Research Notes
If only Mindjet's MindManager could handle LaTeX it would be the perfect tool for organizing my notes, unfortunately it can't and their advice is to use stuff like Equation Editor. That's not for me.
Perhaps at a subconscious level I find that the notes I produce aren't worth saving, fact is that I still haven't found a way of organizing my study notes that suits me. I regularly search the web for tools and today I landed on this MathOverflow question. From this page I jumped to Dror BarNatan's Academic Pensieve, which I invite you to visit because it's unique, impressive and might give you some ideas in organizing your own set of study ( or raw research ) notes.
I have tried a Zillion mindmappers but none come close to MindManager. At work I use FreePlane because it is free ( ( That accurately describes my position with that company ) , the alternative would be to struggle with the tools provided like the damned Word. FreePlane does support LaTeX however.
My little search this morning landed me on DocEar. It supposedly is the MindMapper for Scientists, while MindManager is more positioned at creative business people. If it is any good, I will report on it in a future post.
UPDATE:
I am giving DocEar a try. It's based on FreePlane and JabRef tools that have proved themselves. More later.
Perhaps at a subconscious level I find that the notes I produce aren't worth saving, fact is that I still haven't found a way of organizing my study notes that suits me. I regularly search the web for tools and today I landed on this MathOverflow question. From this page I jumped to Dror BarNatan's Academic Pensieve, which I invite you to visit because it's unique, impressive and might give you some ideas in organizing your own set of study ( or raw research ) notes.
I have tried a Zillion mindmappers but none come close to MindManager. At work I use FreePlane because it is free ( ( That accurately describes my position with that company ) , the alternative would be to struggle with the tools provided like the damned Word. FreePlane does support LaTeX however.
My little search this morning landed me on DocEar. It supposedly is the MindMapper for Scientists, while MindManager is more positioned at creative business people. If it is any good, I will report on it in a future post.
UPDATE:
I am giving DocEar a try. It's based on FreePlane and JabRef tools that have proved themselves. More later.
Sunday, November 24, 2013
Open University: TMA Cutoff date
As an OU student you have to make several TMAs for each course you do. TMA stands for Tutor Marked Assignment. A TMA consists of assignments covering all the booklets you studied in the period prior to the cutoff date of the TMA. The cutoff date is the latest date the work has to be received by your Tutor. This is a period of cutoff dates. Usually you'll find a lot of blog, forum, twitter or facebook posts about TMAs that have been done. Completing a TMA is just part of the study process but for an OU student it's somewhat of an event. Not like an exam, but sending it out gives a feeling of relief, achievement perhaps. Completing a TMA is not something most people do in a few hours, some TMAs take weeks if not months to complete.
The average result of the TMA ( after some formula has been applied to it ), is the maximum result you can get from the course because the final result of the course is the minimum of the exam result and the average TMA score. Also, a minimum TMA score is required to be eligible for taking the exam. For example.
TMA result: 15/100 not eligible for exam.
TMA result: 65/100. Exam result: 100/100. Course result 65/100.
TMA result: 100/100. Exam result: 15/100. Course result 15/100 and thus a FAIL.
Completing all the TMAs on time, requires regular study, and regular study enhances the chance on a good exam result significantly. I suppose that's the thought behind it all. The second example may seem rather undesirable, but it just is not a realistic scenario. A student with a 100% exam score usually has no problems with the TMAs.
Personally, the TMAs are no longer my 'major math challenge'. Slowly but steady I am working on my own mathematical projects. I still need to study, of course, but I am an OU student mainly to justify the time I spend on mathematics to the other stakeholders in my time.  When you say you study mathematics as a hobby, people accept it ( at best ), but explaining that, in fact, you are involved in your own mathematical research is worse than telling that you apply the tools of Scientology in your life. So... I am an OU Student, if you know what I mean. ;)
The average result of the TMA ( after some formula has been applied to it ), is the maximum result you can get from the course because the final result of the course is the minimum of the exam result and the average TMA score. Also, a minimum TMA score is required to be eligible for taking the exam. For example.
TMA result: 15/100 not eligible for exam.
TMA result: 65/100. Exam result: 100/100. Course result 65/100.
TMA result: 100/100. Exam result: 15/100. Course result 15/100 and thus a FAIL.
Completing all the TMAs on time, requires regular study, and regular study enhances the chance on a good exam result significantly. I suppose that's the thought behind it all. The second example may seem rather undesirable, but it just is not a realistic scenario. A student with a 100% exam score usually has no problems with the TMAs.
Personally, the TMAs are no longer my 'major math challenge'. Slowly but steady I am working on my own mathematical projects. I still need to study, of course, but I am an OU student mainly to justify the time I spend on mathematics to the other stakeholders in my time.  When you say you study mathematics as a hobby, people accept it ( at best ), but explaining that, in fact, you are involved in your own mathematical research is worse than telling that you apply the tools of Scientology in your life. So... I am an OU Student, if you know what I mean. ;)
Closed PolyLine  (2)
I developed an algorithm that takes sequences like $\{ 6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4 \}$ and turns them into drawings like this ( in Mathematica, of course ).
In order to determine if the tile above tiles the plane I need to remove all internal lines...
... which turned out a tad more dificult than expected. I got as far as a prototype algorithm which I am currently testing and improving.
To be continued.
Sunday, November 17, 2013
Closed PolyLine
The picture ( below) contains two drawings ( created with Mathematica ), the drawing on the left consists of a hexagon, a square, a triangle and again a square. In order to facilitate an algorithm that decides if this drawing tiles the plane I need a closed polyline to determine if a point is within the borders of the drawing. Oddly enough it is more difficult to create the drawing on the right than the original on the left.
The drawing on the right is a closed polyline, it consists of a number (11) of line segments and has no begin and endpoint, it is closed. ( In another project I am working on we call PolyLines, MultiLines but I haven't seen that name in use elsewhere. )
Geometric cell division
Take a square and add a similar square to one of its sides and remove the shared edge. What remains is a rectangle.
Take an octagon and add a similar octagon to one of its sides and remove the shared edge. What remains is the following polygon.
Take an ngon and add a similar ngon to one of its sides and remove the shared edge. At infinity the ngons will divide into two circles. Unfortunately my computer is too slow to effectively record this in a video.
Take an octagon and add a similar octagon to one of its sides and remove the shared edge. What remains is the following polygon.
Take an ngon and add a similar ngon to one of its sides and remove the shared edge. At infinity the ngons will divide into two circles. Unfortunately my computer is too slow to effectively record this in a video.
n=12 
n=32 
n=64 
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Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before
(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)